Consistency Test

8/9/2019 Consistency Test

1/4

February 1948 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y 345

y,,

7 2 = activity coefficients

Vl, Vz

= molal volumes

of

the liquid components

V ,

=

molal volume in the gaseous stat e

B

= second virial coefficient

Po,

T o = critical pressure and temperature

T =

an arbitrarily fixed temperature

s

=

slope factor, Equations

11

and 12

Y

= relative volatility

LITERATURE CITED

1)

American Petroleum Institute Research Project

44,

National

Bureau of Standards, Selected Valuesof Properties of Hydro-

carbons, Tables 2k (March 31, 944) and 5k (June 30, 1944).

2)

Beatty,

H.

,, nd Calingaert, G., IND. NG.CHEH.,26, 04,

04

1934).

3)

Benedict, M.,ohnson, C. A,, Solomon, E., and Rubin, C.

C.,

4)

Brunjes, A. S., nd Bogart, M.

J.

P.,

IND.

NG.CHEM.,

5, 55

5)

Lewis,G. ., and Randall, M., Thermodynamics, New

York,

(6)

Othmer, D .

F.,

and Benenati, R. F., IND. NQ.CHEM.,7,299

7)Scatchard,

G.,

nd Raymond, C. L.,

J .

Am Chem. SOC.,

60,

278.

8)

Tongberg, C.

O.,

and Johnson,

F.,

IND. NG.CHEM., 5, 733

9)York, R., and Holmes, R. C.,

Ib i d . ,

34, 45 1942).

RECEJVED

ovember 26,1946.

Trans.

Am Inst.

Chem.

Engrs. , 41,

71 1945).

1943).

McGraw-Hill Book Co.,

1923.

1945).

1938).

1933).

Thermodynamics of Nonelectrolyte Solutions)

ALGEBRAIC REPRESENTATION OF THERMODYNAMIC

PROPERTIES AND

THE

CLASSIFICATION

OF

SOLUTIONS

OTTO REDLICH

AND

A.

T.

KISTER

Shell Development Company, Emeryville, Calif.

T h e utilization

of

aboratory data for the design

of

distillation columns and other separation equip-

ment requires the efficient representation of extensive experimental data. A flexible, nonarbitrary,

and convenient method is developed for systems

of

two or more components. This method furnishes

an imm ediate distinction between various types

of

solutions.

XPERIMENTAL results can be improved or damaged on

E heir way from the laboratory to the practical application.

in plant design and operation. The treatmen t of experimental

data should eliminate inconsistencies without distorting the

results by imposing arbitrary conditions, it should be flexible

enough to cover all important cases, and it should be pimple in

operation, From this viewpoint the following method was

developed for the representation of thermodynamic properties

of nonelectrolyte solutions. The present discussion start s from

binary solutions and is later extended to systems of more com-

ponents.

SELECTION OF

A

USEFUL FUNCTION

The use of the activity coefficientsy1 and

y 2

is not advisable

The use of two functions necessitates

This

For this reaeon

Dividing this

1)

since they are redundant.

the imposition of a condition-namely, Duhems equation.

complicates any correct smoothing procedure.

Scatchards excess free energy is preferable.

function by

2.303RT,

one obtains

Q

=

x log YI + (1 - Z og -i2

(z

mole fraction of th e first component)

,

which y a y be considered

a little more convenient in some numerical calculations.

For various reasons still more suitable is the function

dQ/dx = log ( Y I ~ Y ~ ( 2 )

The degree

of this

function is one unit lower tha n Q

as

well

as

the

functions

log

71 = Q

+

(1 - ) d Q / d x

and

log

y~ = Q

- xdQ/dx (3)

This is

a

considerable advantage in practical calculations,

realized in a special case already by Benedict et al. 1). Another

advantage of the function log y1 /y2) is its simple relation to the

experimental data and to the technically important relative

volatility:

(4)

5 )

where

g is

the mole fraction in the vapor and p i and

p i

are the

vapor pressures of the pure components, the vapor being 118-

sumed to be perfect.

Furthermore, the function log -y~/r~)rovides an efficient

tool for eliminating inconsistencies in the experimental data.

Since , according to Equation

1,

assumes the value zero for

x = 0 and x = 1,we derive from Equation 2

a =

y 1

- z ) / x ( l

- y)

log a

=

log

Y l l r n )

-I- log PYP,)

for

Figure 1 shows

a

simple example for the application of this condi-

tion. The only curve for log (rl/r3which represents the da ta in

accordance with Equation

6

is the iero line-that is, the system

is perfect within the limits of experimental error. The deviations

for low concentrations

of

either component are safely recognized

t o be due to experimental errors. One could hardly arrive

so

quickly and cogently

at

the same result by another method.

The relation

7)

following from Equations 5 and

6

is sometimes useful. If the

values of (Y r:fer to a constant temperature, th e integral

is

equal

to log

( p l / p z ) .

If Y is derived from equilibrium measurements

over a moderate temperature interval, the integral can be easily

estimated since usually varies only little with the tempera-

ture. Equation 7 s based only on the assumptions that the

vapor is perfect and that the dependence of the activity coeffi-

cients on the temperature may be neglected.


2/4

I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y

voi.

40, No.

Figure 1. 2,2,4-Trimethylpentan~Methylcyclohex-

ane

a t 741 Mm.

Measurements of Harrison and Berg (3).

Zero line represent.

perfect solution.

S E R I E S DEVELOPMENT

The development of functions like

Q,

log y, etc., into power

aeries with respect to

x

has been suggested already by hlargules.

Por

nonelectrolytes it is always permissible and at least in nu-

merous cases useful. The usefulness, however, depends greatly

on the form of the series. Kohl ( 7 ) , n reviewing the various

methods of representing activity coefficients, pointed out that the

series should be developed in such a way that the higher. terms are

oorrections of the terms of lower order. In this

way

he avoids

the inconvenience of having the result appear

as

a small difference

of

large terms.

Also

important is the fact that the coefficients

of a properly chosen series furnish a natural classification of

various systems, as will be s h o m later.

Since Q = 0 for x = 0 and

x

=

1,

each term must contain the

factor

x ( 1

-

). It

is desirable to develop the series with respect

to a variable which is somehow symmetric with respect to the

two

components. The simplest variable of this kind is 22 - 1, which

merely changes it s sign on exchange of the components. Thus

the most useful development appears to be

Q = ~ ( ls ) [ B + C ( 2 2

-

1) + D ( 2 2 1) +

1 ( 8 )

The coefficients are determined either by plotting

or, preferably, from a diagram of

I

+ D(l - 2.r)ll -

8r l

z ]

+

,

. .

10)

TYPES

OF

S O L U T I O N S

TYPE1.

TYPE

.

The simplest case is the perfect solution for which

The next type, characterized by B 0; C = D =

. .

,

log(r,/rz)

= 0.

= 0, is represented by a straight line in a diagram of log(yl/r2)

against x This straight line must, according to Equat,ions 6

o r

10, pass through zero at

x

= 0.5.

TYPE .

The type

B

0; C 0 ; D = . . . = 0 corresponds

to what is frequently called t'he equation of ?vlargules. If a large

number of experimental points of extremely high accuracy is to be

represented, the method of least squares with properly chosen

weights will be adopted for the determination of the constants

B

and C. In general, however, the following method furnishes

in a few minutes the best results.

The method consists of plotting the experimental data for

log(yl/yz) against 2 drawing a preliminary curve, and reading the

values of log(yl/yz) for a number

of

characteristic points listed in

Tab leI . For type 3-that

is, D

= &only points 1, 3, 5 , 7, and

9

need be considered.

In points 3 and 7 the value of log(yl/yJ is independent

of

C.

In addition, the values of t he function in these two point,s are

equal if three terms are sufficient in Equation 10. This condition

may be assumed to be satisfied in all cases except for extreme

deviations from the perfect solution-that is, especially for in-

completely miscible systems. The equa lity of the values of

log(yl/rt) for p0int.s3 and 7 constitutes, therefore, in many cases,

a

check of the experimental data. The value of

B

- D / 3 , or,

if

D = 0, of

B,

is calculated from the best value of log(yl/y2) for

point,s 3 and

7.

The value

of C

is derived from point 5 .

If

D =

0,

the qui tnt i -

ties B

-

C) and ( - B

-

C) must represent reasonable values for

points 1 and 9. ew curve is drawn through the points

1,

3, 5 ,

7, and 9 calculated according t o Table I with the assumed values

of

B

and C.

The deviations of the experimental points from this curve are

to be judged with regard to Equation 6-that is, only such varia-

tions of the representative curve are permissible which d o not

change the total area under the curve. With a little practico one

sees immediately whether a slight adjustment of B and C will im-

prove the agreement.

The type B 0, D 0, and c = 0 actual1,y exists.

Several systems of methanol and hydrocarbons can

bo

approxi-

mately represented by a corresponding function.

The diagram of

1og(y1/y2) indicates this type immediately by the S-shape of the

representative curve. The absolute values of the function at,z =

0 and x = 1 , are equal, and the curve passes through zero a,f,x =

0.5. The values of B and D are easily derived from points 2, 4,6,

and 8 (Table I).

TYPE , If none of the constants

B ,

C, and

D

equals zero, C is

calculated from point

5

(Table I), B from points 2 and 8, and

D

from point,s 4 and 6 in which the value of t'he 11 term reaches a

maximum.

TYPE.

DISCUSSION

OF

TYPES

A principal advantage of a classification like that discussed in

the preceding section lies in its close connection with the nature of

the solutions and of the components.

It is well known from the work of Hildebrand, Scatchard,

Guggenheim, and others that type 2 very well approximates

systems the components of which are not associated, inte ract

only moderately, and have approximately equal molal volumes.

For systems which do not satisfy the last condition, Scatchard

( 5 )

suggested the relation (Equation

11).

TABLE. CHARACTERISTICOINTS

h-0. 1 2 3 5

0

0 .1464 0.2113 0,2969 0 . 5

f

-

;

D

0.7071B C/4 0,5773

B -

D/3) 0.4082 B -

1)/3)

+

C 4 c/2

N O . 8 7 6 . . /

1 0.8536 0.7887 0.7041 . . .

fog

y I / y I )

- -

C

-

D -0.7071B

-

C 4 -0.5773 B

- D / a ) -0.4082

B

- 0 / 3 ) + C 4

. . .


3/4

February

1948

I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y 341

2.303RTQ

=

AViVzX(1 - z)/[xVI

+

(1 - x VZ]

11)

where Vi and V2 are the molal volumes of t he components. If

this equation is developed into a series similar to Equation

10,

th e comparison of the coefficients furnishes

B

= 2AViV2/2.303RT(Vl f Vz

C

D

=

B(Vi

-

VdZ/(V1

+

Vz)

-

B Vi

- Vz)/(Vi + Vz)

(12)

The D term contributes less than 0.011B to log(yl/yz) if the

larger molal volume exceeds the smaller by less than 50 .

Scatchard's equation therefore can be replaced in this case for

data of moderate accuracy by a formula of type

3

with the

value

of

C given in Equat ion

12,

This formula is usually more

convenient than the original equation.

Figure 2 illustrates ?n application of this formula. Scatchard's

equation has been found to approximate well all hydrocarbon

eystems except those containing benzene.

It will be shown in another paper of this series th at association

of one of the components produces

a

contribution to log(-yi/rz)

which has approximately the character of the

ll

term in Equation

10. Types 4 and 5 therefore will be discussed in connection with

association. A few results, however, should be antic ipated here.

Interassociation between the two components tends to diminish

the influence of the term characteristic of association. This ex-

plains why systems of highly associating substances often belong

to type 3.

The association term is maintained if both components associate

but not with each other, Apparently the system benaene-

cyclohexane belongs to this type.

According to Hildebrand

(4 )

both substances are slightly associated. The measurements of

Scatchard, Wood, and Mochel(6) can be represented by Equat ion

10 with a finite value of D.

It should be pointed o ut, however, thab the existqnce of this

small D term could be ascertained only by measurements of high

accuracy. Within the limits of error of the usual technique the

system would belong to type

3

rather than type 5. Thus the

classification presented here furnishes approximations of various

degrees. This flexibility may be considered to be an advantage of

a power series.

MULTICOMPONENT SYSTEMS

The series used for binary systems can be extended without

The definition of the

ifficulty to multicomponent systems.

function Q by Equation 1 s now replaced by

Q =

X Z ~

og ~k

(13)

The relation of this function with the free energy furnishes for the

activity coefficient

The differentiation is to be performed

at

constant pressure and

temperature. Also, all mole fractions are to be kept constant

except the one indicated in the differential quotient.

It is convenient for some calculations t o represent Q as a sum

of terms

Q h

which are homogeneous of the degree h in the mole

fractions xk Equation 14 can then be written as

(15)

og

y r

= Z B Q h / a X r - ( h - ) Q h '

The functions

log

(rr/y*)

= a Q / a ~ r- aQ/axa

(16)

offer the same advantages as the function defined in Equation 2.

Their degree is one unit lower than that of 6? or logy?,and they are

closely connected with the relative volatility.

The development of

Q

nto

a

power series can be appropriateIy

generalized without difficulty. It is useful to rewrite Equation 8

in the more symmetric form

Qlz

= ZIXZ[BIZ

Clz(~1

-

2,)

+ DIZ(ZI

-

Z ) ~+

. .

I

17)

The series for a ternary system is then conveniently represented

by

Q Q23 +

Q3i

+ Q i z

+

2 1 2 2 2 3

[ C

+

D i(~z

-

3 )

D2(23

-

21)

+

. .

I

(181

The number of coefficients required for a series running to a

certain degree has been discussed by Benedict

et

al. ( I ) . The

present representation offers again the advantage of a natu ral

classification by means

of

terms

of

decreasing importance.

Equations

16,

17, nd

18

furnish for a ternary system

log ( 7 z / 7 i ) = Blz(21

-

2) + Cm[3(21- 2 z ) *

-

11/2

+ DlZ(Z1- XZ)[(Xl

-

2 ) 2 - 4x12221 + . . .

+ 23[B23

-

B81 f Cz3(222 - 3 + c31(221

-

3 )

+ Dz8 32s2- 4x228 + 3 ) - D31(32l2- 42123 + 2 3 )

+ c ( Z l - 2 ) + D1(-2321 2x122 +

22x3

- ZZ)

+

02 +23zl + 22122 - 22x3

-

i z ) f s . I

19)

The actual calculation is simple, since the higher te rms are always

small and often negligible. Cyclic permutat ion of the subscripts

furnishes log( 3 / 7 2 ) and log(y~/y ,~).

For

a system of four components Q is represented by the sum of

the six binary functions of Equation

17,

the four terms containing

21x223, etc., and a term containing X I X Z X ~ X P The further gen-

eralization is obvious.

EXAMPLE:HEPTANE-METHANOL-TOLUENE.he measure-

ments of Benedict

et

al.

I )

of vapor-liquid equilibria in the

system

n-heptane-methanol-toluene

furnish an opportunity to

show that even accurate measurements in a ternary system can be

sufficiently well represented by the functions Q

for

the binary

systems alone, provided that adequate expressions for these

systems have been found.

Systems with an associating component like methanol are

properly represented by means of the association function

A ( K , x ) = 22 log (1

+

K )

-

210g (1 - 22

+

11 + 4 K ~ ( l ~) ] )2 ) / 2 (1

- 2)

(20)

0.1

t o

0.05

t 0

m

-

-

0.05

0.1

1

0 x

0 . 5

Figure 2. Toluene-2,2,4-Trimethylpentane at

760

M m .

Menmaremento

of

Driakamar,

Brown,

and White 3).

Curve

repremntm

Santahard a

equation.


4/4

348

I N D U S T R I A L

A N D

E N G I N E E R I N G C H E M I S T R Y

Vol.

40, No. 2

coefficients-namely, eight from the binary jystems and three

TABLE1. RELATIVE OLATILITY

F

~ - H E PT A N E - ~ ~ E T H A N O Lerived from the ternary system. The practical importance

of

the restriction of the number of empirical coefficients need not be

strrssed

A T 760

AIM.

RIethanol/Heptane

1 oc. x

Obsvd.

Benediet This paper

CONCLUSIBSS

60.60 0 .138 16.1 16.45 16. 5

59.47 0.178 12.6 12.65 12.55

58.93 0,390 4.4 3 4.45 4.36

58.82 0.668 1.46 1.49 1.47 The use of the funct'ion log( rl/rz ) for the representation of

0.810 0.696 0.691 0.702

free energy da ta of binary solutions is advisable. A suita-

8.81

59.01 0 ,885 0,423 0,407 0.419

59.90 0.946 0,242 0,243 0.246 bly chosen power series represents this function in a flexible and

unbiased ~iay.

The coefficients can be derived quickly from

TABLE

11.

t ,

oc.

70.25

66.44

65.58

64.47

64.10

63.79

63.67

63.58

63.62

63.94

RELATIVE

OLATILITYF >\IETHBSOL-TOLCENF:

AT

760 MM.

RIethanol/Toluene

22 Obsvd. Benedict

This

paper

0.130

0.266

0,407

0.593

0.692

0.779

0 ,843

0,852

0.927

0,969

19.0

9.90

5 , 9 4

3.10

2.16

1 .54

1 .23

1 .01

0 . 81

0 .61

15.07

9.75

5.72

3

07

2.17

1 . 6 5

1.19

0 .99

0.80

0.62

1 7 , s

9 . 8 3

5.93

3.13

2.20

1 .58

1 . 2 1

1 . 0 1

0.80

0.81

rxperimental data.

The nature of various classes of solutions is expressed in the

magnitude of these coefficients. The behavior of log(

y , / y 2 )

therefore can be predicted to a certain degree. Scatchard's

cquation furnishes

a

good approximation for systems of hydro-

carbons except those containing benzene.

The series used for the free energy can be advantageously used

also for other propertie., such as heat content and entropy.

ACKNOWLCDGIIENT

The

authors cxpress their

gratitude to It. W. Millar and

G. Muller for helpful discus-

sions on t,he subject

of

this and

the preceding paper. L. Korba

assisted the authors in the

-

~

.

TABLE

1.

RELATIVE

70L.ITILITICY

O F

n - I l c P . r . ~ ~ ~ - - 1 ~ ~ T H , 4 ~ O L - - T n L U ~ S E4T

760 lIA>I,

Heptanc/Toluene IIcthanol/Toluene

t , ae.

El

I2 O h s v d . Bencdict This paper O b s v d . Benedict Thiq paper

90 .77

0.0684 0,8486

2.75

2.78 2 .72 1 . 15 1 .19

1.24

69 .96

o.1.51.0 0.7733 2.11

2.26

2.32 1.58 1 .64 1 .80 calculations.

61.59

0 .0 7 8 7

0,7494

2 .58 2 . 6 3 2 .48

1 .91 1 .77

1 . 8 8

59.97 0.3412

0.5409

1 .65

1 .8

1 .83

3 . 8 3 . 9

4 .20

m61.35 0.1 717 0.603 8 2 . 0 7 2.17 2 .08 3 .22 3 .25 3 .26

LITERATURE CITED

82.01 0.0911 0,6114

2.43

2.42 2.25 3.13

3 . 0 6

3 . 0 8

64.34

0.0962

0,3711 2.24 2 . 1 3 2.04

7.14

6 . 8

6 .92

62.96

0.2533 0,3172

1 , 8 5

1.92 1 .82

8 . 3 8 . 5

8 . 6

(1)

Benedict,

M . ,

Johnson, C.

A.,

Solomon, E., and

Rubin, L. C., Trans. Am.

Inst. Chem.

Engrs.,

41 ,

371 (1945).

( 2 ) Drickarner, H. G . , B r o w n ,

G. G., and

IVhite, R . R., Ibid.,

4) Hildehraild, J. H.,

J . them. p h y s . , 7,

234 (1939).

In tkis equation the mole fract,ion z refers to the associating

The three binary systems have been found to be represented

b v

41,

55 (1945).

component,

and

is a Onstant characteristic Of this component'

(3)

Rarrison, J . M , , andBerg,

L,,

IND,

st CHEM.,

8, 117 (1946).

means of the coefficients

(1

= heptane, . 2 = methanol, 3 = (5) Scatchard, G., Chem. Revs. , 8, 321 (1931).

toluene)

(6) Scatchard,

G . ,

JVood.

S.

E., and

M o o h e l ,

J. M., .

P h ~ s .

hem.,

7) W o h l , K . , T,.ans. Am,. h t .

C hem.

E i i g i s . , 42,

215

(1946).

43, 119 (1930).

B23

= 0.95;

Bai

= 0.117;

Biz

= 1.178

Kz

=

3.8;

c3

=

0.018; D 2

= 0.155

(21)

R E C E I V E D

November

2 5 , 1946

ail o ther coefficients being zero. Accordingly,

the followingequations were used for the ternary

system:

log (?1/? 3) =

0 2 2 8 ~ ~O . 1 1 7 x 3 xi) + 0 O l 8 [ z , ( ~ ~2s3)

(23)

The corrections

for

the imperfection of

the

vapor

calculated by Benedlct et

at .

ue ie used.

The results are compared n ith the experinlental

and calculated values of BenediLt

e l

al in Tables

11,111, and IV. These tables contain the tem-

perature, t h e

mole

fractions in the Irqurd, and the

relative volatilities observed and calculated

by

Bcnerhct et al. and by means of Equations

22

and

23.

The volatilities calculated

by

mean3

01

these

equations agree with the observed values ap-

proximately a5 well

as

the values calculated by

Benedict et al.; but Equations 22 and 23 con-

tain only six coefficients,

all

derived

from

the

binary systems. Bonedict et d ntroduced eleven

k

Za Z8 - 2Z1)]

+

0.155~2(3~:

Z l X z + 2:)

Furfural Solvent Extraction Unit for

Lube Oil

Consistency Test

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